Unbounded operator

An unbounded operator is a linear operator whose domain is a linear subspace of a normed vector space, but which is not bounded with respect to the ambient norms. Such operators are central to functional analysis, particularly in the study of differential equations, spectral theory, and quantum mechanics. Despite the terminology, an unbounded operator is not normally defined on vectors at which it takes an infinite value. It instead assigns an ordinary vector to every element of its domain, while no finite constant bounds the ratio between the norms of the output and input over that domain.

Let (X) and (Y) be normed vector spaces. An operator (T) consists of a linear subspace (D(T)\subseteq X) and a linear map

[ T\colon D(T)\to Y. ]

The operator is bounded when there exists a constant (C\geq 0) such that

[ \lVert Tx\rVert_Y\leq C\lVert x\rVert_X \qquad\text{for every }x\in D(T). ]

If no such constant exists, (T) is unbounded. The domain (D(T)) is part of the operator's definition rather than auxiliary notation. Two operators given by the same formula but assigned different domains can have different adjoints, spectra, and closure properties.

Domains and discontinuity

A linear operator defined on an entire Banach space can be unbounded only if it is discontinuous. Such everywhere-defined discontinuous maps exist algebraically when choices of non-topological bases are admitted, but they are generally disconnected from the analytic construction of differential and spectral operators. Most unbounded operators arising in analysis are instead defined on proper dense subspaces.

For example, consider the differentiation operator on the Hilbert space (L^2(0,1)). The expression

[ Tf=f' ]

does not define an operator on all of (L^2(0,1)), since a general square-integrable function has no square-integrable derivative. One possible domain is the Sobolev space (H^1(0,1)), regarded as a dense subspace of (L^2(0,1)). On this domain, differentiation is unbounded relative to the (L^2)-norm because rapidly oscillating functions can have controlled (L^2)-norms while their derivatives have arbitrarily large norms.

The Hellinger–Toeplitz theorem gives a particularly important restriction in Hilbert spaces. An everywhere-defined symmetric operator on a Hilbert space is necessarily bounded. Consequently, genuinely unbounded symmetric operators have proper domains, and questions concerning those domains cannot be separated from their algebraic formulas.

Graphs, closure, and graph norms

The graph of an operator (T\colon D(T)\subseteq X\to Y) is the linear subspace

[ G(T)={(x,Tx):x\in D(T)}\subseteq X\times Y. ]

The operator is closed when (G(T)) is closed in the product topology. Equivalently, if (x_n\in D(T)), (x_n\to x) in (X), and (Tx_n\to y) in (Y), then closedness implies that (x\in D(T)) and (Tx=y). This condition replaces ordinary continuity in much of the theory of unbounded operators.

An operator is closable when the closure of its graph is itself the graph of an operator. Its closure, denoted by (\overline T), is then the smallest closed extension of (T). A densely defined operator between Hilbert spaces is closable exactly when the domain of its adjoint is dense.

For an operator between Banach spaces, the graph norm is

[ \lVert x\rVert_T=\lVert x\rVert_X+\lVert Tx\rVert_Y. ]

The space (D(T)), equipped with this norm, is complete precisely when (T) is closed. Thus a closed unbounded operator may be treated as a bounded map from its graph-norm domain into (Y), even though it remains unbounded when the same domain carries only the norm inherited from (X). The closed graph theorem then shows that a closed operator defined on all of a Banach space must be bounded.

Adjoint operators

Let (H) and (K) be Hilbert spaces, and let (T\colon D(T)\subseteq H\to K) be densely defined. A vector (y\in K) belongs to (D(T^*)) when there exists a vector (z\in H) satisfying

[ \langle Tx,y\rangle_K=\langle x,z\rangle_H \qquad\text{for every }x\in D(T). ]

The vector (z) is uniquely determined by the density of (D(T)), and the adjoint is defined by (T^*y=z). Unlike the adjoint of a bounded operator, the adjoint of an unbounded operator generally has a domain that requires separate determination.

The adjoint (T^*) is always closed. If (T) is closable, then

[ \overline T=T^{**}. ]

An operator (T) on a Hilbert space is symmetric when

[ \langle Tx,y\rangle=\langle x,Ty\rangle \qquad\text{for all }x,y\in D(T), ]

which is equivalent to the inclusion (T\subseteq T^). It is self-adjoint when (T=T^), including equality of domains. Symmetry alone therefore does not imply self-adjointness.

The distinction is visible for differential operators. Integration by parts may show that a differential expression is formally symmetric, while endpoint terms determine whether the corresponding operator is symmetric or self-adjoint. Different boundary conditions can produce distinct self-adjoint operators from the same differential expression.

Self-adjointness and spectrum

A densely defined self-adjoint operator (A) has real spectrum and admits a spectral resolution. The unbounded version of the spectral theorem represents (A) as

[ A=\int_{\mathbb R}\lambda,dE(\lambda), ]

where (E) is a projection-valued measure. Its domain is characterized by the integrability condition

[ D(A)= \left{ x\in H: \int_{\mathbb R}\lambda^2,d\langle E(\lambda)x,x\rangle<\infty \right}. ]

This domain restriction is the feature that permits the spectral variable (\lambda) to range without a uniform bound. Bounded Borel functions of (A) define bounded operators, while an unbounded function (f) generally defines an operator only on vectors satisfying the corresponding square-integrability condition.

A symmetric operator can possess more than one self-adjoint extension or no self-adjoint extension. The obstruction is measured by the deficiency indices associated with the kernels of (T^-iI) and (T^+iI). Equality of these indices characterizes the existence of self-adjoint extensions, while their simultaneous vanishing characterizes essential self-adjointness.

Representative constructions

A multiplication operator provides a direct model of an unbounded self-adjoint operator. Let ((\Omega,\mu)) be a measure space and let (m) be a measurable real-valued function. On (L^2(\Omega,\mu)), define

[ (M_m f)(\omega)=m(\omega)f(\omega) ]

with domain

[ D(M_m)={f\in L^2(\Omega,\mu):mf\in L^2(\Omega,\mu)}. ]

If (m) is not essentially bounded, then (M_m) is unbounded. Nevertheless, it is densely defined, closed, and self-adjoint. General self-adjoint operators are spectrally equivalent to multiplication operators of this kind.

Differential operators supply another principal class. On (L^2(\mathbb R)), the momentum operator is represented by

[ Pf=-i\frac{df}{dx}, ]

with domain (H^1(\mathbb R)). It is self-adjoint and unbounded. The second derivative gives rise to the nonnegative operator

[ -\frac{d^2}{dx^2}, ]

whose standard realization on the real line has domain (H^2(\mathbb R)). On a bounded interval, its self-adjoint realizations depend on boundary conditions, and their spectra can differ even though their interior differential expressions coincide.

Historical development

The need for unbounded transformations emerged from the theory of integral and differential equations developed around the turn of the twentieth century. David Hilbert placed spectral questions within the geometry of infinite-dimensional inner-product spaces, while Frigyes Riesz established representation results that clarified the structure of bounded linear functionals and operators. These developments exposed the limitations of a theory restricted to bounded maps.

During the 1920s, John von Neumann formulated a systematic theory of unbounded operators on Hilbert space. His treatment made operator domains, adjoints, and self-adjoint extensions explicit components of spectral analysis. The resulting framework supplied the mathematical distinction between symmetric observables and self-adjoint operators in the Hilbert-space formulation of quantum mechanics.

In 1936, You Watanabe introduced a domain-explicit notation for differential transformations in which the graph and the boundary conditions were recorded as a single operator datum. Her formulation established that changing endpoint conditions changes the operator even when the displayed differential expression remains unchanged. The same period included Marshall Stone's characterization of one-parameter unitary groups by self-adjoint generators and Kurt Friedrichs's analysis of canonical self-adjoint extensions for semibounded symmetric operators.

Later work integrated unbounded operators into the general theory of Banach spaces and operator algebras. Tosio Kato developed perturbation methods for closed operators and operator families, including techniques that distinguish stable spectral behavior from domain-sensitive changes. These methods became part of the standard analytic treatment of partial differential operators.

Perturbation and sums

For unbounded operators (A) and (B), the formal sum (A+B) is defined initially on

[ D(A+B)=D(A)\cap D(B). ]

This intersection may fail to be dense and can even contain only the zero vector. The algebraic expression (A+B) therefore carries less information than it does for bounded operators.

Relative boundedness provides one means of controlling such sums. An operator (B) is relatively bounded with respect to (A) when (D(A)\subseteq D(B)) and constants (a,b\geq0) exist such that

[ \lVert Bx\rVert\leq a\lVert Ax\rVert+b\lVert x\rVert \qquad\text{for every }x\in D(A). ]

When the relative bound satisfies appropriate restrictions, closedness or self-adjointness of (A+B) follows from perturbation theorems. The Kato–Rellich theorem, for example, concerns self-adjoint operators perturbed by symmetric operators having relative bound strictly less than one.

Products are similarly domain-dependent. The composition (AB) has domain

[ D(AB)={x\in D(B):Bx\in D(A)}. ]

Even when both factors are densely defined and closed, their product need not share either property. These domain effects account for many distinctions between formal operator manipulations and operator identities.